Corollary 11.5.0.22. See Remark 3.5.1.19.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Proposition 3.1.7.1, we can choose a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r] \ar [d]^{f} & X' \ar [d]^{f'} \\ Y \ar [r] & Y' } \]
where the horizontal maps are inner anodyne, $Y'$ is a Kan complex, and $f'$ is a Kan fibration. Then $f$ is $n$-connective if and only if $f'$ is $n$-connective, and $f$ is a weak homotopy equivalence if and only if $f'$ is a homotopy equivalence (Proposition 3.1.6.13). The desired result now follows by applying Proposition 3.2.7.2 to the Kan fibration $f'$. $\square$